Consider a system (a gas) of point-like particles with a gravitation-like interaction (potential) $V(r) \sim \frac{1}{r}$ between pairs of them.

One can rule out statistically that two particles will approach each other exactly along their line, so two particles will never collide directly - thus giving rise to infinite (kinetic) energies. So the particles will always whirl around each other in one way or the other.

What can be said about such a system in the framework of statistical mechanics?

How does its partition function look like?

What are its thermodynamical equilibrium states?

What happens when cooling such a system down (e.g. from very high temperatures)?

  • $\begingroup$ I might be wrong, but wouldn't such a system be able to provide an infinite amount of energy upon cooling? $\endgroup$ – DumpsterDoofus Apr 1 '14 at 0:06
  • $\begingroup$ @DumpsterDoofus: You probably are right - but that's the kind of objection I am looking for! (But maybe the system is all that there is - and it cannot be cooled "from the outside"?) $\endgroup$ – Hans-Peter Stricker Apr 1 '14 at 0:16
  • $\begingroup$ I'm not sure exactly how to tackle this because I would have to consider interactions between every pair.. but why wouldn't they all be at r=0 in equilibrium? The Boltzmann factor for r=0 between all particles would be infinite for this configuration wouldn't it? $\endgroup$ – Julien Apr 1 '14 at 0:29
  • $\begingroup$ I believe this is a major unsolved problem in cosmology, although I haven't looked into it in any depth. $\endgroup$ – Nathaniel Apr 1 '14 at 1:48
  • $\begingroup$ I think at about $V^{1/3}= \frac{2V \rho G}{c^2}$ it would form a black hole. $\endgroup$ – Mirc Breitschuh Apr 1 '14 at 9:14

A (3d) gas of particles with a gravitational interaction is an example of a system with long range interactions, where the energy is not additive and thus many basic results of classical statistical mechanics are not valid, including the equivalence of the microcanonical, canonical and grand-canonical ensembles. For a general introduction to the subject see these lecture notes by Mukamel, and for an introduction to the stat. mech. of gravitational systems see these lecture notes by Padmanabhan. Because ensembles are not equivalent, one must be careful when discussing issues such as the paritition function of the system.

I am not sure why you say that particle collisions can be rules out. As far as I understand, a short length-scale cutoff (i.e., a minimum distance between particles) must be introduced to avoid a collapse of particles into the same point. Even with such a cutoff, I believe that particles will phase-separate into what is called a core and a halo: the core is a compact and rather still aggregate of particles; because its potential energy is very low (i.e., large and negative), it enables the existance of a "hot" halo with very high kinetic energy. If I am not mistaken, this halo will spread out towards infinity, and thus to have a sensible equilibrium state one must also introduce a large distance cutoff (Padmanabhan's lecture notes discuss this issue, but I've only briefly refreshsed my memory now so my answer may be slightly inaccurate). In the discussion above I've assumed a microcanonical system.

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